Nonextensive Thermodynamics of the Morse Oscillator: Signature and Solid State Application

TL;DR

This paper explores the thermodynamics of the Morse oscillator within Tsallis nonextensive statistics, revealing significant effects at low temperatures and extending the analysis to solid-state applications, highlighting nonextensive influences on vibrational properties.

Contribution

It provides an analytical derivation of the generalized partition function and thermodynamic quantities for the Morse oscillator under nonextensive statistics, including novel insights into low-temperature behavior.

Findings

Nonextensive framework restricts accessible states.

Generalized internal energy and entropy depend on temperature and q.

Schottky-type anomaly observed in specific heat.

Abstract

In this work, we present a detailed thermodynamic analysis of a bound quantum system: the Morse oscillator within the framework of Tsallis nonextensive statistics. Using the property of the bound spectrum (upper bound) of the Morse potential, limited by the bond dissociation energy, we analytically derive the generalized partition function. We present results for both the high- and low-temperature limits. We propose the effective number of accessible states as a measure of nonextensivity. The calculation shows that the nonextensive framework further restricts the number of accessible states. We also derive the generalized internal energy and entropy and examine their dependence on temperature and the nonextensivity parameter \( q \). Numerical results confirm the strong effect of nonextensive behavior in the low-temperature regime (precisely low to moderate temperature), where the ratio of generalized internal energy and internal energy calculated from the Boltzmann Gibbs (BG) formula develops a nontrivial dip structure for \( q < 1 \). Moreover, the generalized specific heat shows the Schottky-type anomaly. We extend our study by deriving the specific heat of solids with BG and Tsallis statistics using the anharmonic energy levels of the Morse oscillator. This study suggests that the Morse oscillator is a solvable and physically meaningful testing ground for exploring the thermodynamics of quantum systems driven by nonextensive statistics, with implications for the vibrational properties of the non-equilibrium molecular thermodynamics (especially diatomic molecules).